Question

Solve the equation. Help please!
`x^{log_(3)x-4 } = 3^(-3) \\\\x^{log_(2)x } = 16`

Answer 1

First equation:

`x^(\log_3x-4)=3^(-3)`

`\log_3\left(x^(\log_3x-4)\right)=\log_33^(-3)`

`(\log_3x-4)\log_3x=-3\log_33`

`(\log_3x)^2-4\log_3x=-3`

`(\log_3x)^2-4\log_3x+3=0`

`(\log_3x-3)(\log_3x-1)=0`

`\log_3x-3=0\text{ or }\log_3x-1=0`

`\log_3x=3\text{ or }\log_3x=1`

`3^(\log_3x)=3^3\text{ or }3^(\log_3x)=3^1`

`x=27\text{ or }x=3`

Second equation:

`x^(\log_2x)=16`

`x^(\log_2x)=2^4`

`\log_2\left(x^(\log_2x)\right)=\log_22^4`

`\log_2x\log_2x=4\log_22`

`(\log_2x)^2=4`

`\log_2x=\pm2`

`2^(\log_2x)=2^(\pm2)`

`x=2^2\text{ or }x=2^(-2)`

`\boxed{x=4\text{ or }x=\frac14}`